Optimal Noise Benefit in Composite Hypothesis Testing under Different Criteria

The detectability for a noise-enhanced composite hypothesis testing problem according to different criteria is studied. In this work, the noise-enhanced detection problem is formulated as a noise-enhanced classical Neyman-Pearson (NP), Max-min, or restricted NP problem when the prior information is completely known, completely unknown, or partially known, respectively. Next, the detection performances are compared and the feasible range of the constraint on the minimum detection probability is discussed. Under certain conditions, the noise-enhanced restricted NP problem is equivalent to a noise-enhanced classical NP problem with modified prior distribution. Furthermore, the corresponding theorems and algorithms are given to search the optimal additive noise in the restricted NP framework. In addition, the relationship between the optimal noise-enhanced average detection probability and the constraint on the minimum detection probability is explored. Finally, numerical examples and simulations are provided to illustrate the theoretical results.